A stable Nyström interpolant for some Mellin convolution equations

We consider the numerical solution of second kind integral equations of the form $$u(y) - \int\limits_0^1 {k(y/x)\frac{{u(x)}}{x}dx = f(y), 0 \le y \le 1,} $$ for some given kernelk(t). These equations, usually indicated as of Mellin type, arise in a variety of applications. In particular, we examine a Nyström interpolant based on the following product quadrature rule: $$\int\limits_0^1 {k(y/x)\frac{{u(x)}}{x}dx \approx \sum\limits_{i = 0}^n {w_{ni} (y)u(x_{mi} ).} } $$ This rule is obtained by interpolatingu(x) by the Lagrange polynomial associated with the set of Gauss-Radau nodes {x _ni}. Under certain assumptions on the kernelk(t), we are able to prove the stability of our interpolant and derive convergence estimates.     

Lelong numbers of positive plurisubharmonic currents

A (k,k)-current T on an open subset of C^N is plurisubharmonic if is positive. Positive plurisubharmonic currents admit Lelong numbers; we prove here that they are independent on the coordinates system. Moreover, if Y is an analytic subset of pure dimension, then the Lelong numbers of T on Y are given by a non negative weakly plurisubharmonic function.     

The actions ofS n+1 andS n on the cohomology ring of a coxeter arrangement of typeA n−1

Let be the complexified Coxeter arrangement of hyperplanes of typeA _n−1. In this paper we construct anS _n+1 extension of the naturalS _n action on the complex cohomology ring of the complement ofA _n−1. Recurrence formulas connecting characters with respect to theS _n and theS _n+1 action are given.     

Di-2-pyridylketone semicarbazone as ligand in metal complexes: synthesis and X-ray crystal structure

A series of complexes of di-2-pyridylketone semicarbazone (Hdips) and Mn(II), Co(II), Co(III), Ni(II) and Cu(II) nitrates were synthesized and characterized by means of IR spectroscopy and for cobalt and nickel by X-ray crystal structures. The results are in agreement with the formulae: Mn(Hdips)₂(NO₃)₂·2H₂O, [Co(Hdips)₂](NO₃)₂·H₂O (I), [Ni(Hdips)₂](NO₃)₂·H₂O (II), Cu(Hdips)(NO₃)₂·2H₂O, [Co(dips)₂](NO₃)·2H₂O (III). The structure of I and II are monoclinic, space groupP2₁/c, with, I,a=15.980(4),b=11.531(2),c=16.170(2)Å;=104.20(2)°,Z=4,R=0.032; II,a=16.109(5),b=11.480(3),c=16.135(6)Å;=104.15(2)°,Z=4,R=0.069. Compound III is also monoclinic, space groupP2₁/c witha=12.173(5),b=15.619(5),c=15.338(8)Å;=111.40(4)°,Z=4,R=0.059. In these complexes the ligand is tridentate via carbonylic oxygen, semicarbazone and pyridine nitrogens forming each two five membered chelate rings with the metal in a distorted octahedral geometry.     

Intermolecular proton transfer in solid phase: a rare example of crystal-to-crystal transformation from hydroxo- to oxo-bridged iron(III) molecule-based magnet

Intermolecular proton transfer in solid phase from the hydroxo bridge to a water molecule occurs in a new mu-hydroxo iron(III) compound of formula {EtNH3[Fe2(ox)2Cl2(mu-OH)].2H2O}n leading to a still crystalline compound in which the mu-oxo bridge replaces the mu-hydroxo one. Both three-dimensional compounds exhibit magnetic ordering at Tc ca. 70 K due to a spin canting.     

Gadolinium(III) Complexes of dota‐Derived N‐Sulfonylacetamides (H4(dota‐NHSO2R)=10‐{2‐[(R)sulfonylamino]‐2‐oxoethyl}‐1,4,7,10‐tetraazacyclododecane‐1,4,7‐triacetic Acid): A New Class of Relaxation Agents for Magnetic Resonance Imaging Applications

Four new ligands for lanthanide ions based on the H₃do3a (=1,4,7,10‐tetraazacyclododecane‐1,4,7‐triacetic acid) structure and bearing one N‐sulfonylacetamide arm were synthesized, i.e., H₄dota‐NHSO₂R=10‐{2‐[(R)sulfonylamino]‐2‐oxoethyl}‐1,4,7,10‐tetraazacyclododecane‐1,4,7‐triacetic acids 1a – e . A ¹⁵N‐NMR study of the ¹⁵N‐labelled Eu³⁺ complex of one such ligands, 1d , showed that the coordination of the N‐sulfonylacetamide arm involves the carbonyl O‐atom rather than the N‐atom. The relaxometric properties of the corresponding Gd³⁺ complexes were investigated as a function of pH and temperature. These complexes have relaxivities in the range 4.5–5.3 mM ^?1 s^?1, at 20 MHz and 25°, and are characterized by a single H₂O molecule in their inner coordination sphere. The mean residence lifetime of this molecule is relatively long (500–700 ns) compared to other anionic complexes. The slow rate of H₂O exchange can be justified by the extensive delocalization of the negative charge on the N‐sulfonylacetamide arm. The long residence time of the coordinated H₂O allowed the observation of the effect of the prototropic exchange on the relaxivity. The study of the interaction between the complex [Gd( 1e )]‐ and HSA revealed a weak affinity constant highlighting the importance of a localized negative charge on the complex to promote a strong interaction with the protein.     

N,O-heterocyclics. 13. Isoxazolidinium salts as synthons to N,N, O-trisubstituted hydroxylamines

Substituted isoxazolidinium salts react with lithium aluminium hydride to yield open-ring products which have hydroxylamine structures. The bimolecular reaction-mechanism has been investigated by substituent effect and the structure of the products ascertained by spectroscopic methods with the aid of the MIKE technique. The overall process of the ring-opening substitution is controlled by the polarisation of the C? N bond with steric and conformational factors acting mainly at the C-5 position of the nucleus. The mechanism of isoxazolidinium ion reaction defines the use of these synthons towards the synthesis of N,N,O-trisubstituted hydroxylamines and substituted 1,3-amino-alcohols.     

Asymmetric synthesis of 1-substituted-1-(pyridin-2-yl)methylamines by diastereoselective reduction of enantiopure N-p-toluenesulfinyl ketimines

Reduction of enantiopure N-p-toluenesulfinyl ketimines derived from 2-pyridyl ketones affords N-p-toluenesulfinyl amines with good yields and diastereoselectivities.